### Poster session

**Mariusz Budziński**(University of Warsaw)

**Quantum families of quantum group homomorphisms**
We define a quantum family of homomorphisms of Hopf algebras. Roughly
speaking, we show that such a family is classical. Moreover, we show
that a quantum family of homomorphisms of Hopf algebras is consistent
with the counits and coinverses of the given Hopf algebras.
**Antoine Caradot**(Institut Camille Jordan)

**Inhomogeneous Kleinian singularities and quivers**
Starting with a finite group of SU_2(C), we construct the Kleinian singularities linked to the Dynkin diagrams of type A, D, E and explained some known results about them. Then, using symmetries on Dynkin diagrams, we give a definition due to P. Slodowy of singularities linked to the inhomogeneous Dynkin diagrams. The notions of deformation and resolution can be extended to the inhomogeneous Kleinian singularities. H. Cassens and P. Slodowy constructed the semi-universal deformation of a Kleinian singularity of type A, D or E using symplectic geometry and quiver theory. We will present how we can extend their construction to the inhomogeneous Kleinian singularities and the consequences of some computations.
**Szymon Charzyński**(University of Warsaw)

**Wei-Norman equations for classical and exceptional groups**
We show, that the non-linear autonomous Wei-Norman equations, expressing the solution of a linear system of non-autonomous equations on a Lie algebra, can be reduced to the hierarchy of matrix Riccati equations in the case of all classical simple Lie algebras. The result cannot be fully extended to all simple Lie algebras, in particular to the exceptional G2 algebra.
**Tomasz Czyżycki**(Institute of Mathematics, University of Bialystok)

**Generalized discrete orbit function transforms of affine Weyl groups**
The four types of orbit functions are a generalization of the standard symmetric and antisymmetric orbit sums. They can also be viewed as multidimensional generalizations of one-dimensional cosine and sine functions with the symmetry and periodicity determined by the affine Weyl groups.
We classify two independent admissible shifts, which preserve the symmetries of
the weight and the dual weight lattices and generalize the form of orbit functions.
The discrete Fourier calculus and discrete orthogonality relations of the generalized orbit functions are formulated.
**Diana Dziewa-Dawidczyk**(Warsaw University of Life Sciences - SGGW)

**Jacobi polynomials and representations of group $SU(2)$ (together with prof. Aleksander Strasburger)****Agota Figula**(Institute of Mathematics, University of Debrecen)

**Multiplicative loops of topological quasifields**
An interesting application of Lie theory in geometry is the area of the coordinatization of topological non-desarguesian affine planes. Locally compact connected topological non-desarguesian translation planes have been a popular subject of geometrical research since the seventies of the last century (cf. [1]). These planes are coordinatized by locally compact quasifields $Q$ such that the kernel of $Q$ is either the field $\mathbb R$ of real numbers or the field
$\mathbb C$ of complex numbers. The multiplicative loop $Q^{\ast }=
(Q \backslash \{ 0 \}, \cdot )$ is homeomorphic to $\mathbb R \times S^n$, where $S^n$ is the $n$-sphere with
$n \in \{1,3,7\}$. The group $G$ topologically generated by the left translations $\lambda_a$, $a \in Q^{\ast }$ is a linear Lie group, namely a closed connected subgroup of the group $GL_{n+1}(\mathbb R)$ if the kernel of $Q$ is $\mathbb R$, whereas a closed connected subgroup of the group $GL_{2}(\mathbb C)$ if the kernel of $Q$ is $\mathbb C$. In the talk we wish to determine the algebraic structure of the multiplicative loops for topological connected quasifields and describe explicitly the quasifields which coordinatize locally compact translation planes admitting a large Lie group as collineation group.
**Elitza Hristova**(Institute of Mathematics and Informatics, Bulgarian Academy of Sciences)

**Invariants of the symplectic and the orthogonal groups acting on $GL(n)$-modules**
Let $W$ be a polynomial $GL(n,\mathbb{C})$-module. Then $W$ can be written as a direct
sum $W=\oplus_\lambda k(\lambda)V^n_\lambda$, where $\lambda= (\lambda_10,\hdots,\lambda_n)\in \mathbb{Z}^n_{\geq 0}$ is a partition and $V_\lambda^n$ is the
irreducible $GL(n,\mathbb{C})$-module with highest weight $\lambda$. We consider the algebra of
polynomial functions $\mathbb{C}[W]$ and the algebras of invariants $\mathbb{C}[W]^G$ where $G$ is one of
$SL(n,\mathbb{C})$, $O(n,\mathbb{C})$, and $Sp(2k,\mathbb{C})$ (in the case $n = 2k$). In the paper [BBDGK] the authors develop a method for computing the Hilbert series $H(\mathbb{C}[W]^{SL(n,\mathbb{C})}, t)$ of the
algebra of invariants $\mathbb{C}[W]^{SL(n,\mathbb{C})}$. Instead of the Molien-Weyl integral formula, this
method uses the theory of rational symmetric functions. In our poster we discuss
the method from [BBDGK] and show how to extend it to compute also the Hilbert
series of the algebras of invariants $\mathbb{C}[W]^G$ for $G = O(n,\mathbb{C})$ and $G = Sp(2k,\mathbb{C})$.
We give explicit examples for computing $H(\mathbb{C}[W]^G, t)$, some of which were known
and some of which were not known before. We discuss also further generalizations
and applications of this method. The poster is based on a joint work with Vesselin
Drensky.**Katarzyna Karnas**(Center for Theoretical Physics, Polish Academy of Sciences)

**Criterions for quantum gates unversality**

Slides from this talk
The conditions for a set of quantum k-mode unitary and orthogonal gates to be universal are presented. It is shown that if the spectra of considered gates do not belong to a finite list of exceptional spectra, the problem of deciding universality boils down to solving a set of linear equations. We also present how the exceptional spectra are determined and prove that their number grows exponentially with the number of modes. Finally, for 2- and 3-mode gates it turns out that the exceptional spectra correspond to either finite subgroups of SU(2) (SO(3)), or give universality. This way we classify all universal pairs of one qubit gates.
**Anna Kimaczyńska**(Lodz University)

**Elliptic boundary conditions in the symmetric bundle**
A complete set of elliptic boundary conditiobs for second order $O(n)$-invariant
differntial operators in the bundle of symmetric tensors will be presented.
**Arthemy Kiselev**(University of Groningen)

**Deformation approach to quantisation of field models**
(based on the IHES/M/15/13) Associativity-preserving deformation quantisation $x \rightarrow \star$ via the Kontsevich summation over weighted graphs is lifted from the
algebras of functions on finite-dimensional Poisson manifolds to the
algebras of local functionals within the variational Poisson geometry
of gauge fields over the space-time.
**Wasyl Kowalczuk, Barbara Gołubowska, Ewa Eliza Rożko**(Institute of Fundamental Technological Research of the Polish Academy of Sciences)

**Towards affine dynamical symmetry in mechanics of deformable bodies**
Certain dynamical models of deformable bodies, including problems of
partial separability and integrability, are discussed. There are some
reasons to expect that the suggested models are dynamically viable and
that on the fundamental level of physical phenomena the
“large” affine symmetry of dynamical laws is more justified
and desirable than the restricted invariance under isometries.
**Abdelmalek Mohammed**(Abubker Belkaid University, Tlemcen)

**Transversality of special manifolds in arbitary codimensions****Giovanni Moreno**(Institute of Mathematics, Polish Academy of Sciences)

**A representation-theoretic characterisation of completely exceptional second-order PDEs**
The class of "completely exceptional PDEs" was introduced by Peter Lax in 1954. He studied the phenomenon of development of shock waves out of discontinuity waves, realising that, for the solutions of certain equations, such a transition never occur after a finite time. He called these equations "completely exceptional", since they behave like linear PDEs, though being genuinely nonlinear. In 1991 Guy Boillat proved that, by integrating Lax’s conditions of complete exceptionality, one obtains precisely the Monge-Ampere equations. In this talk, based on a joint publication with J. Gutt and G. Manno (doi:10.1016/j.geomphys.2016.04.021) I will show that completely exceptional second-order PDEs can be parametrised by the elements of the kernel of a natural differential operator. In order to prove the existence such an operator, I will introduce a suitable geometric framework for nonlinear second-order PDEs, and use some elementary representation-theoretic techniques. Our operator allows to describe intrinsically the original "completely exceptionality conditions" found by Lax, but also to introduce more general (quadratic, cubic, etc.) Monge-Ampere equations.
**Natalie Nikitin**(University of Paderborn)

**Exponential laws for weighted function spaces and regularity of weighted mapping**
Let $U\subseteq\mathbb{R}^n$, $V\subseteq\mathbb{R}^m$ be open subsets and $E$ be a locally convex topological
vector space. For $k,l\in\mathbb{N}_0\cup\{\infty\}$ a function $\gamma:U\times V\to E$ is called $C^{k,l}$ if the maps
$\frac{\partial^\alpha}{\partial x^\alpha}\frac{\partial^\beta}{\partial y^\beta}\gamma:U\times V\to E, (x,y)\mapsto \frac{\partial^\alpha}{\partial x^\alpha}\frac{\partial^\beta}{\partial y^\beta}\gamma(x,y)$
$\frac{\partial^\alpha}{\partial x^\alpha}\frac{\partial^\beta}{\partial y^\beta}\gamma:U\times V\to E, (x,y)\mapsto \frac{\partial^\alpha}{\partial x^\alpha}\frac{\partial^\beta}{\partial x^\beta}\gamma(x,y)$
are defined and continuous for all $\alpha\in\mathbb{N}_0^n$, $\beta\in\mathbb{N}_0^m$ with $|\alpha|\leq k$, $|\beta|\leq l$.
For natural topologies on the function spaces, an exponential law of the form $C^{k,l}(U\times V,E)\cong C^k(U,C^l(V,E))$
was shown by H. Alzaareer and A. Schmeding ([1]). We prove analogous exponential laws for weighted spaces of
$C^{k,l}$-maps ([2]), and also for spaces of $C^{k,l}$-maps on suitable topological groups, like metrizable or locally compact
groups ([3]). As an application, certain Lie groups of weighted mappings are shown to be $C^k$-regular. Exponential
laws for function spaces related to infinite-dimensional Lie groups have also been established by A. Kriegl, P. W. Michor
and A. Rainer ([4]) in a special case of Schwartz spaces of vector-valued rapidly decreasing smooth functions. Our
results concerning topological groups improve findings by D. Belti\c{t}\u{a} and M. Nicolae ([5]).
**Antoni Pierzchalski**(Lodz University)

**Natural elliptic boundary conditions for different geometries**
A short review of complete lists of natural boundary conditions for different geometries like Riemannian,
Hermitaian, symplectic etc. will be given. The problem of their ellipticity in the sense of Gilkey-Smith
will be mentioned. The significance of the branching rule for subgroups of suitable Lie groups will be noticed.
**Xavier Rivas**(Polytechnic University of Catalonia)

**A constraint algorithm for k-precosympletic field theories and some applications****Farshid Saeedi**(Islamic Azad University, Mashhad, Iran)

**On the dimension of the Schur multiplier of $n$-Lie algebras**
For an $n$-Lie algebra $A$ of dimension $d$, we find the upper bound $\dim \M(A) \leq\binom{d}{n}$, where $\M(A)$ denotes the multiplier of $A$ and that the equality holds if and only if $A$ is abelian. Finally, we give a formula for dimension of the multiplier of direct sum of two $n$-Lie algebras.
**Daniel Wysocki**(University of Warsaw)

**Structure and classification of Lie bialgebras**
We study the problem of determination and classification of non-equivalent Lie bialgebra structures on a fixed Lie algebra. We introduce invariants on classes of Lie bialgebras giving the same Lie algebra structure on the dual Lie algebra. This is employed to simplify the differentiation of non-equivalent Lie bialgebra structures. Special attention is paid to coboundary Lie bialgebras. The classification of Lie bialgebras on $\mathfrak{sl}(2,\mathbb{R})$ and $\mathfrak{su}(2)$ is studied in detail.

*References*:

[1] Salzmann, H., Betten, D., Grundhöfer, T., Hähl, H., Löwen, R., Stroppel, M. (1995).

*Compact projective planes*, Walter de Gruyter.

*References*:

[BBDGK] F. Benanti, S. Boumova, V. Drensky, G.K. Genov, P. Koev,

*Computing with rational symmetric functions and applications to invariant theory and PI-algebras*, Serdica Math. J. 38 (2012), Nos 1-3, 137-188.

__:__

**Best poster**These are the results of the joint work with Adam Sawicki.

*References*:

[1] Alzaareer, H., Schmeding, A.,

*Differentiable mappings on products with different degrees of differentiability in the two factors*, Expo. Math. 33, No. 2 (2015), 184-222

[2] Nikitin, N.,

*Exponential laws for weighted function spaces and regularity of weighted mapping groups*, preprint, arXiv:1512.07211

[3] Nikitin, N.,

*Exponential laws for spaces of differentiable functions on topological groups*, in preparation

[4] Kriegl A., Michor P.W., Rainer A.,

*The exponential law for spaces of test functions and diffeomorphism groups*, Indag. Math. (N.S.) 27, No. 1 (2016), 225–265

[5] Belti\c{t}\u{a}, D., Nicolae, M.,

*On universal enveloping algebras in a topological setting*, Stud. Math. 230, No. 1 (2015), 1-29